Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys

2978 V. Vaithyanathan et al. / Acta Materialia 52 (2004) 2973–2987
energetics is applicable only for substitutional fcc-based
configurations. It can be used for calculating the free
energy of an fcc-based compound (e.g., a structure that
can be represented by a substitutional decoration of A
and B atoms on the sites of an fcc lattice, such as an
ordered L1 2 or L1 0 ), but cannot be used for calculating
the free energy of h 0 , which has a (distorted) CaF 2
crystal structure. Hence, the h 0 energy is obtained from
direct first-principles calculations at T ¼ 0 K, coupled
with the vibrational entropy of h 0 , which has been found
to be unexpectedly important for this phase [1]. Additionally,
by performing point defect calculations of supercells
of h 0 and inserting the energetics into a lowtemperature
expansion, we find that the configurational
entropy of h 0 is small, and we do not consider h 0 offstoichiometry.
The h 0 formation enthalpy was calculated
from first-principles as )195.8 meV/atom at Al 2 Cu
stoichiometry (X Cu ¼ 1=3). The computed (formation)
vibrational entropy is )0.62k B [1]. Hence, the free energy
of h 0 as a function of temperature is given by
DF h
0ðT Þ¼ 195:8 þ 0:62k B T ðin meV=atomÞ: ð15Þ
We note that without the inclusion of the h 0 vibrational
entropy term, the calculated equilibrium solubility of Cu
in solid solution is extremely small. For easier visualization,
the free energy is scaled by a linear term in
composition ½ 3X Cu DF h
0ðT ÞŠ, such that the rescaled h 0
free energy at X Cu ¼ 1=3 is zero (Fig. 4(c)). This linear
scaling of the free energy does not affect the determination
of equilibrium composition through the common
tangent construction.
3.2. Interfacial energy
In a system where the precipitate and matrix phases
are substitutional rearrangements of atoms on the same
lattice type, and form perfectly coherent interfaces (e.g.,
GP zone phases), one could calculate the interfacial free
energy from a Monte Carlo simulation coupled with
thermodynamic integration, analogous to the procedure
described for bulk free energies. Alternatively, T ¼ 0K
values may be obtained from direct first-principles supercell
calculations (without using a CE). h 0 precipitates
are partially coherent with different crystal structures for
precipitate and matrix. Hence, it is not amenable to the
CE method. Therefore, we extract the T ¼ 0 K interfacial
energies directly from first-principles supercell calculations.
We next describe how the interfacial energies
are extracted from supercell energetics, and separated
from the coherency strain energetics.
We begin by considering an N-atom coherent supercell
containing an interface between two materials, A
and B (in the case of this work, A ¼ Al and B ¼ Al 2 Cu
h 0 ). For simplicity, we consider the case of cells comprising
equal amount of A and B. The energy of a such
an A N=2 B N=2 supercell can be separated into two components
[27]: (a) coherency strain (cs): the strain energy
required to maintain coherency between the (lattice
mismatched) materials A and B, and (b) interfacial energy:
the energy associated with the interactions between
materials at the A=B interface(s). To define these terms,
it is useful to first consider the infinite period supercell
limit N !1, for a supercell with interface along an
orientation ^G with lattice constants a k and a ? parallel
and perpendicular to ^G, respectively. In this infiniteperiod
case, A=B interfacial interactions (which scale as
the area of the interface) contribute a negligible amount
to the supercell formation energy dE sup (which scales as
the volume of the superlattice):
dE sup ðN !1; ^GÞ
1
dE CS ð^GÞ ¼min
a ?
2 dEepi A
ða ?; a A k ; ^GÞ
þ 1
2 dEepi B ða ?; a B k ; ^GÞ ;
ð16Þ
where dE sup is the energy of the supercell relative to
equilvalent amounts of A and B in their equilibrium bulk
geometries. In Eq. (16), the materials A and B are deformed
in an ‘‘epitaxial’’ geometry: Both materials are
brought to a common lattice constant a ? perpendicular
to ^G, and the energy of each material is individually
minimized with respect to the lattice constant a k parallel
to ^G. The epitaxial energies dE epi are the energies of A
and B in these epitaxial geometries relative to their
equilibrium bulk energy. For finite-period supercells, the
energy is determined not only by the coherency stain
energy, but also by the interfacial energy r times the
number (2) and area (A) of these interfaces. Since we are
using energetics per atom, we must divide by the number
of atoms in the cell, N. The interfacial energy is then
defined as
dE sup ðN; ^GÞ dE sup ðN !1; bGÞ 2rð^GÞA : ð17Þ
N
Combining Eqs. (16) and (17), we see the decomposition
of the supercell energy (for any period) into strain and
interfacial components
dE SL ðN; ^GÞ ¼ 2rð^GÞA þ dE CS ð^GÞ:
ð18Þ
N
From Eq. (18), we see that if the supercell formation
energies (per atom) are plotted as a function of 1=N, the
slope is just 2rA, and the y-intercept is dE CS ð^GÞ. We
have extracted the interfacial energies from first-principles
supercell calculations using the construction of Eq.
(18). The results are shown in Fig. 6. We note that extracting
the coherency strain energy from these calculations
is not numerically stable; a more robust
approach is the direct calculation of Eq. (16) described
below.
The tetragonal structure of h 0 precipitate embedded in
an fcc matrix results in partially coherent plate-shaped
precipitates. The h 0 plates possess a broad coherent face